[Math] Assume I’m unhappy with the Riemann Rearrangement Theorem. How else can I define a series

convergence-divergencedefinitionreal-analysissequences-and-series

I have been explaining the Riemann Rearrangement Theorem to a friend of mine, and they feel as though the definition of series using partial sums "doesn't work" for conditionally convergent sequences. I understand how they feel: the RRT feels counter-intuitive, and so the result should be denied as a contradiction and the partial-sum definition rejected.

However, the definition of a series as

$$\sum_{k=1}^{\infty} a_k = \lim_{n\to\infty} \left(\sum_{k=1}^n a_k\right)$$

feels like the most natural approach to take. If I wanted to add infinitely many numbers together by hand, this is how I would have to do it.

Is there an alternative (but not equivalent) definition of an infinite sum that agrees with the partial-sum definition on absolutely convergent sequences, but where the RRT doesn't hold?

My guess is that if you define some value for an infinite series agreeing on absolutely convergent sequences then this method of definition must imply the RRT, but I don't see how that proof could go.

Best Answer

Assume that we want a notion of summability $\sum'$ such that

  1. (Compatibility with absolutely convergent series) $$\sum_{n\geq 1}|a_n|<+\infty\quad\Longrightarrow\quad\sum_{n\geq 1}'|a_n|=\sum_{n\geq 1}|a_n|$$
  2. (Compatibility with the vector space operations and other reasonable assumptions) Provided that both $\sum'a_n$ and $\sum' b_n$ are finite, $\sum'(a_n+\lambda b_n)=\sum' a_n+\lambda\sum' b_n$. Additionally $$ \left|\sum' a_n\right|<+\infty\quad\Longrightarrow\quad \lim_{n\to +\infty}a_n=0,$$ $$ a_n>0,\left|\sum' a_n\right|<+\infty\quad\Longrightarrow\quad \sum' a_n>0$$
  3. (Negation of Riemann-Dini) Provided that $\sum_{n\geq 0}'a_n$ is finite, for any bijective $\sigma:\mathbb{N}\to\mathbb{N}$ $$ \sum_{n\geq 0}'a_n = \sum_{n\geq 0}'a_{\sigma(n)}$$

Then such notion of summability is precisely the notion of absolute-summability. Assume that $\sum'a_n$ is finite and $\{a_n\}$ has an infinite number of both positive and negative terms. By $3.$ we may assume without loss of generality that the sign of $a_n$ agrees with the parity of $n$. If $\sum a_n$ is not absolutely convergent then $\left|\sum a_{2n}\right|$ or $\left|\sum a_{2n+1}\right|$ is unbounded (or both are). By 2. and 3. both $\sum' a_{2n}$ and $\sum' a_{2n+1}$ have to be finite. Assuming that $|\sum a_{2n}|$ is unbounded, by 2. and 3. again $$ \sum_{n\geq 1}'|a_{2n}|\geq \sum_{n=1}^{N}|a_{2n}| $$ has to hold for any $N\in\mathbb{N}^+$, but that implies $\left|\sum' a_{2n}\right|=+\infty.$

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